Exploiting NOMA in D2D Assisted Full-duplex Cooperative Relaying

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER Exploiting NOMA in D2D Assisted Full-duplex Cooperative Relaying Tu-Trinh THI NGUYEN , Dinh-Thuan DO Department of Electronics and Telecommunications, Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, 12 Nguyen Van Bao, 700000 Ho Chi Minh City, Vietnam nguyen[email protected], dodinhth[email protected] DOI: 10.15598/aeee.v19i3.4116 Article history: Received Feb 20, 2021; Revised Jul 09, 2021; Accepted Jul 20, 2021; Published Sep 30, 2021. This is an open access article under the BY-CC license. Abstract. In a wireless system, dual-hop transmission requires Full-Duplex (FD) to transmit signals from the base station too far users. It is more beneficial if we deploy non-orthogonal multiple access to serve specific users, i.e. normal users (near and far users) and device-to-device users. The fairness and outage performance of these users can be studied. We particularly focus on mathematical analysis of outage performance which is computed based on Signal to Noise Ratio (SNR) of received signals at each kind of user. We derive a closed-form formula of such outage probability along with throughput. To realize both the FD NOMA, this paper performs system performance metrics and considers how self-interference make influences system performance. The simulation results validate the theoretical analysis and show that our scheme can obtain a better outage probability and throughput performance with high transmit SNR at the base station and lower required target rates. Keywords NOMA, outage probability, throughput. 1. Introduction Considering as a prominent approach for the increasing requirements for higher capacity in wireless systems, Non-Orthogonal Multiple Access (NOMA) has recently emerged in the upcoming Fifth-Generation (5G) wireless communications [1], [2], [3], [4], and [5]. To eliminate multi-user interference, the conventional multiple access, namely Orthogonal Multiple Access (OMA), employs orthogonal allocation of resources. The OMA scheme has some kinds, including Time Division Multiple Access (TDMA), Frequency Division Multiple Access (FDMA), and Code Division Multiple Access (CDMA). In principle, multiple users in NOMA can share the same frequency and time resources [6], [7], [8], [9], and [10]. NOMA systems allocate different power levels to different users by changing the level of interference from other users [11]. The wireless systems get benefits from other advances of NOMA systems such as spectral efficiency, low latency, and connectivity which are provided to meet the main requirements of the upcoming 5G wireless communications [12]. However, higher complexity at the receivers using Successive Interference Cancellation (SIC) to eliminate the interference from other users’ signals and then detecting their own signals is enabled. By assigning different power levels to different users, NOMA networks exhibit user fairness based on their channel conditions. In particular, users achieve high power coefficients due to their weak channels, while users with stronger channels are assigned with lower power factors. NOMA with the presence of technologies such as Device-to-Device (D2D) communications provides the heterogeneous nature of 5G cellular systems. The operation of D2D pairs can reuse the spectrum band of cellular users [13], [14], and [15]. The integration of D2D transmission mode into the cellular system makes an interference to broadcast channels or multiple access. In [16] and [17], the application of NOMA in D2D communications has been investigated. In [18], the authors explored the interplay mode as a special D2D approach for the NOMA system. In such a system, the power domain multiplexing is required for both the D2D pair and cellular users to elimi212 ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER |gU1←R|2= Γ  m|gU1←R|2, ξ2α2 λ|gU1←R|2 m|gU1←R|2 +ξ3α4 λ|gU1←R|2 m|gU1←R|2 U1.(1) nate the strong interference between them by the SIC decoding. They presented the case that D2D pair employing the selection between the interplay mode and underlay mode, the SIC decoding constraint is achieved at both D2D receiver and NOMA base station. Reference [19] considered a D2D-enhanced Unmanned Aerial Vehicles (UAV) network, in which D2D is conducted to improve the file dispatching efficiency. Considering such D2D-enhanced UAV systems relying on NOMA, the authors studied graph theory-based algorithms regarding resource allocation. The authors in [20] presented the power allocation sub-problem with continuous variables and decoding order variables. In particular, they first studied a heuristic algorithm to optimize the power allocation for NOMAbased with given D2D power allocation. [21] presented a collaborative protocol with joint power optimization in the D2D-NOMA system. As such, to limit signal leakage, while performing beamforming to suppress AN in the legitimate users’ directions, a FullDuplex (FD) cellular receiver injects the Artificial Noise (AN) signals to deteriorate the eavesdropper’s channel. Other merging D2D-NOMA systems can be reported in [21], [22], [23], [24], [25], [26], [27], [28], [29], and [30]. Motivated by work [31], this paper studies performance of D2D groups, in which the near and far users can operate along with D2D users under the context of the NOMA protocol. 2. System Model In Fig. 1 we consider a downlink NOMA using the dualhop transmission. In such system, the Base Station (BS) intends to send signals to the near user and the far user, i.e. two cellular users U1(near user) and U2 (far user). Especially, U2needs assistance from one full-duplex relay acting user (R) for forwarding signal from BS. In this scenario, a D2D user D1is located in serving coverage of such BS. Due to the blockage and hindrance to signal propagation, we cannot process the signal in direct link BS-to U2while the remaining links of the NOMA system are available. Benefiting Full-Duplex (FD) mode, the relay re-transmits the decoded symbol only to the far user. Contrarily, in such NOMA, Ris able to forward symbol U2 and its own symbol to D1at the same time which is achieved by the enabler of NOMA. In such NOMA system, we examine wireless channels following Nakagamimfading model. In particular, the channel coefficient experiences Nakagami-mfading will be represented as Gamma distribution with integer fading factor mz and mean λzdenoted by |z|2∼Γmz,λz mz. It is noted that relay Rexhibits imperfect SelfInterference (SI) cancellation causing residual at relay |f|2∼Γm|f|2, ξ1 λ(|f|2) m(|f|2)with (0 ≤ξ1≤1). At each hop, power levels are reset, i.e. α1, α2, α3, α4are power allocation coefficients, where α1+α3= 1,α1< α3 and α2+α4= 1,α2< α4.ρB=PB σ2 2 and ρR=PR σ2 2 are considered as transmit Signal to Noise Ratio (SNR) at the BS and R, with PB,PRthe total transmit power of BS and the total transmit power of relay, respectively and σ2 2is the variance of Additive White Gaussian Noise (AWGN) at R. The channel coefficient of interference link from Eq. (1). Fig. 1: System model. Is the level of residual interference (0 ≤ξ3≤1). The PDF and CDF of |gz|2is given by [33]. f|z|2(x) = xmz−1 Γ (mz)βmz z e −x βz,(2) and F|z|2(x) = 1 −e −x βz mz−1 X n=0 xn n!βn z ,(3) where βz ∆ =λz mz with mzand λzrepresent the integer fading factor and the channel mean power λz=E{|z|2}.Γ(.)is the gamma function xis a variable. To further examine system performance metrics, we need to compute Signal-to-Interference-plusNoise Ratio (SINR). Following the principle of NOMA, the relay is able to decode signal x3by considering x1 as noise and the corresponding SINR at Ris formulated ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 213 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER by [31]. γx3 R←BS =α3ρB|gR←BS|2 α1ρB|gR←BS |2+ρR|f|2+ 1.(4) To decode signals x3and x1at U1, we need to determine the received SINRs respectively as: γx3 U1←BS =α3ρB|gU1←BS|2 α1ρB|gU1←BS |2+ρR|gU1←R|2+ 1,(5) and γx1 U1←BS =α1ρB|gU1←BS|2 ρR|gU1←R|2+ 1.(6) At the second hop transmission, the far/weak user U2receives the signal transmitted from BS and decodes it’s information x3by treating x2as noise. Therefore, the corresponding SINR to detect signal x3at U2is given by: γx3 U2←R=α4ρR|gU2←R|2 α2ρR|gU2←R|2+ 1.(7) Besides two users’ signals, D2D user D1needs ro receive the transmitted signal from Rin this second hop transmission. First, D1needs to decode x3if treating x2as noise. By employing SIC, it can be decoded its own signal x2. Hence, we can obtain SINRs that correspond correspond to detect x3and x2at D1respectively as: γx3 D1←R=α4ρR|gD1←R|2 α2ρR|gD1←R|2+ 1,(8) and γx2 D1←R=α2ρR|gD1←R|2.(9) The achievable rates of U1and D1are respectively written by: CU1= log2(1 + γUx1 1←BS),(10) and CD1= log2(1 + γDx2 1←R).(11) Moreover, the achievable rate of U2can be obtained by as: CU2= log2(1 + min(γx3 R←BS , γx3 U1←BS , γx3 U2←R, γx3 D1←R)). (12) Finally, the overall achievable capacity can be calculated as: Cpro. cap. =CU1+CD1+CU2.(13) 3. Outage Probability Analysis In this section, it is necessary to determine an important performance metric, i.e. outage probability. Due to differences in terms of power allocation factor and decoding order, such system performance for each user could be different. The definition of outage probability represents probability to SINR less than the specific thresholds which are decided by different demands of users. We first analyze the outage performance of the near user as follow. 3.1. Outage Probability of U1 Considering the performance metric for the considered system, the Outage Probability (OP) at U1can be explained as: Outage behavior will occur in U1related to two situations. First, if it can not decode the signal x3. Second, if it decodes x3but it cannot decode x1. From the above description, the outage probability of U1is formulated by: PU1= Pr log21 + γx3 U1←BS < R2, log21 + γx1 U1←BS < R1! = Pr γx3 U1←BS < δ2, γx1 U1←BS < δ1 = 1 −Pr γx3 U1←BS ≥δ2, γx1 U1←BS ≥δ1, (14) where the threshold SNRs are δ1= 2R1−1, δ2= 2R2−1. Substituting the formula Eq. (6) and Eq. (7) into formula Eq. (14) we get: PU1= 1 −Pr   A1ρB|gU1←BS |2≥ρR|gU1←R|2+ 1, α1 δ1 ρB|gU1←BS |2≥ρR|gU1←R|2+ 1   (15) where A1=α3−α1δ2 δ2 . The existence of OP reported in (15) is related to situation δ2>α3 α1 , the OP becomes PU∞=∞and for δ2<α3 α1 , the OP can be rewritten by: PU1= 1− ∞ Z0 F|gU1←BS |21 ϕρU2B (ρRx+ 1)f|gU1←R|2(x)dx. (16) From formulas Eq. (2) and Eq. (2) we can calculate F|gU1←BS |21 ϕρB (ρRx+ 1)and f|gU1←R|2(x)as 214 ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER follows: F|gU1←BS |21 ϕρB (ρRx+ 1)= 1 −e −ρRx+ 1 ϕρBβgU1←BS · · mgU1←BS −1 X n=0 1 ϕρB (ρRx+ 1)n n!βn gU1←BS , (17) f|gU1←R|2(x) = xmgU1←R−1 ΓmgU1←R· ·e −x (ξ2α2+ξ3α4)βgU1←R (ξ2α2+ξ3α4)βgU1←RmgU1←R. (18) Substituting Eq. (17) and Eq. (18) into Eq. (16), we get: PU1= 1 − mgU1←BS −1 P n=0 n1 P k=0 n1 ke −1 ϕρBβgU1←BS · ·ρk R n!βn gU1←BS ϕnρn BΓmgU1←R· ·1 ξ2α2βgU1←R+ξ3α4βgU1←RmgU1←R· · ∞ R0 e−µxxk+mgU1←R−1dx, (19) where µ∧ =ρR ϕρBβgU1←BS +1 ξ2α2βgU1←R+ξ3α4βgU1←R and ϕ= min A1,α1 δ1. By using result in [32] and applying some polynomial expansion manipulations, Eq. (19) is computed by: PU1= mgU1←BS −1 P n=0 n1 P k=0 n1 k· ·ρk Rk+mgU1←R−1! n!βn gU1←BS ϕnρn BΓmgU1←R· ·e −1 ϕρBβgU1←BS µ−k−mgU1←R ξ2α2βgU1←R+ξ3α4βgU1←RmgU1←R. (20) 3.2. Outage Probability of U2 If Rfails to decode x3or Rcan decode but U2can not, then outage occurs in U2. Hence, the OP of U2is calculated by: PU2= Pr log2(1 + γx3 R←BS )< R2, log21 + γx3 U2←R< R2! = Pr γx3 R←BS < δ2, γx3 U2←R< δ2 = 1 −Pr γx3 R←BS ≥δ2, γx3 U2←R≥δ2. (21) Replace Eq. (4) and Eq. (7) into Eq. (21), we have: PU2= 1 −Pr        A1ρB|gR←BS |2≥ρR|f|2+ 1,    α4−α2δ2 | {z } ∆ =℘    ρu|gU2←R|2≥δ2        = 1 −Pr A1ρB|gR←BS |2≥ρR|f|2+ 1, ℘ρR|gU2←R|2≥δ2! = 1 −Pr A1ρB|gR←BS |2≥ρR|f|2+ 1 | {z } ∆ =Ψ1 · ·Pr |gU2←R|2≥δ2 ℘ρR | {z } ∆ =Ψ2 . (22) If δ2>α3 α1 and δ2>α4 α2 exist, the OP becomes PU2= 1, whereas for δ2<α3 α1 and δ2<α4 α2 it can be expressed: PU2= 1 −Ψ1Ψ2 = 1 − mgR←BS −1 P n=0 n1 P k=0 mgU2←R−1 P n2=0 n1 k!· ·ρk R(k+mf−1)! n!βn gR←BS 1 ρBA1n1 · ·e −1 ρBA1βgR←BS −δ2 ℘ρRβgU2←R Γ (mf) (ξ1βf)mf· ·ρR ρBA1βgR←BS +1 ξ1βf−k−mf · ·δn2 2 n2!℘ρRβgU2←Rn2, (23) where Ψ1,Ψ2can be calculated as follows: Ψ1 ∆ =P|gR←BS |2≥δ2 ρB(α3−α1δ2)ρR|f|2+ 1 = ∞ R01−F|gR←BS |2ρR ρBA1 x+1 ρBA1· ·f|f|2(x)dx. (24) ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 215 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER From Eq. (2) and Eq. (3) we can calculate F|gR←BS |2ρR ρBA1 x+1 ρBA1and f|f|2(x)as follows: F|gR←BS |2ρR ρBA1 x+1 ρBA1 = 1 −e −ρR ρBA1βR←BS x · ·e −1 ρBA1βR←BS mR←BS −1 P n=0 1 n!βn R←BS · ·ρR ρBA1 x+1 ρBA1n , (25) and f|f|2(x)dx =xmf−1 Γ (mf)βmf f exp −x βf.(26) Based on [32] and from the Eq. (24), Eq. (25), Eq. (26), Ψ1can be calculated as: Ψ1=e −1 ρBA1βgR←BS · · ∞ R0 mgR←BS −1 P n=0 e −ρR ρBA1βgR←BS x n!βn gR←BS · ·ρR ρBA1 x+1 ρBA1nxmf−1e −x ξ1βf Γ (mf) (ξ1βf)mfdx = mgR←BS −1 P n=0 n1 P k=0 n1 k!1 n!βn gR←BS · ·ρR ρBA1k1 ρBA1n1−k e −1 ρBA1βgR←BS · · ∞ R0 e −ρR ρBA1βgR←BS xxkxmf−1 Γ (mf) (ξ1βf)mfe −x ξ1βfdx. (27) Based on [32] and applying some polynomial expansion manipulations, Ψ1is given by: Ψ1= mgR←BS −1 P n=0 n1 P k=0 n1 k!ρk R n!βn gR←BS 1 ρBA1n1 · ·e −1 ρBA1βgR←BS (k+mf−1)! Γ (mf) (ξ1βf)mf· ·ρR ρBA1βgR←BS +1 ξ1βf−k−mf . (28) Similarly, the following result can be achieved: Ψ2 ∆ =P|gU2←R|2≥δ2 ℘ρR =e −δ2 ℘ρRβgU2←R mgU2←R−1 P n2=0 δn2 2 n2!℘ρRβgU2←Rn2. (29) 3.3. Outage Probability of D1 The two situations to D2D user meet outage behavior which is related to conditions: D1fails to decode U2’s signal and D1decodes x3but fails to decode signal x2. Therefore, the OP of D1is computed as: PD1= Pr log21 + γx3 D1←R< R2, log21 + γx2 D1←R< Rd! = Pr γx3 D1←R< δ2, γx2 D1←R< δd = 1 −Pr γx3 D1←R≥δ2, γx2 D1←R≥δd. (30) By replacing Eq. (9) and Eq. (10) into Eq. (30), PD1 can be recomputed by: PD1= 1 −Pr    α4ρR|gD1←R|2 α2ρR|gD1←R|2+ 1 ≥δ2, α2ρR|gD1←R|2≥δd    = 1 −Pr     |gD1←R|2≥δ2 (α4−δ2α2)ρR , |gD1←R|2≥δd α2ρR     = 1 −e −ζ βgD1←R mgD1←R−1 P n=0 ζn n!βn gD1←R (31) where ζ∆ = max δ2 (α4−δ2α2)ρR ,δd α2ρR. 3.4. Throughput Based on Eq. (20), throughput of U1can be given by: TU1= (1 −PU1)R1 =                1− mgU1←BS −1 P n=0 n1 P k=0 n1 k!· ·ρk R(k+m˜g21 −1)! n!βn gU1←BS ϕnρn BΓmgU1←R· ·e −1 ϕρBβgU1←BS µ−k−mgU1←R ξ2α2βgU1←R+ξ3α4βgU1←RmgU1←R                R1 . (32) 216 ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER From Eq. (23), we have: TU2= (1 − PU2)R2 =                          mgR←BS −1 P n=0 n1 P k=0 mgU2←R−1 P n2=0 n1 k!· ·ρk R(k+mf−1)! n!βn gR←BS 1 ρBA1n1 · ·e −1 ρBA1βgR←BS −δ2 ℘ρRβgU2←R Γ (mf) (ξ1βf)mf· ·ρR ρBA1βgR←BS +1 ξ1βf−k−mf · ·δn2 2 n2!℘ρRβgU2←Rn2                          R2. (33) Finally, throughput of D1can be written as: TD1= (1 − PD1)Rd =  e −ζ βgD1←R mgD1←R−1 P n=0 ζn n!βn gD1←R   Rd. (34) 4. Numerical Results In this section, we numerically simulate some theoretical results from some figures to show the outage performance. In particular, main parameters can be seen in Tab. 1. Figure 2 depicts how outage performance can be improved at high transmit SNR ρb=ρB. It can be seen that lower outage probability can be achieved as high SNR. This situation can be explained that high SNR leads to better SINR metrics and then such OP can be enhanced. In the case of fading parameter m= 2, the outage performance of the second user outperforms that of the remaining users. The main reason is that different conditions of decoding and power allocation factors lead to different values of OPs for users. By increasing, the quality of wireless channels, m= 4 is reported as the best case OP. We can confirm the exactness of derived formulas by matching Monte-Carlo and analytical simulations, i.e. such matching is very tight. The OP performance of user U1remains at floor value at high SNR. This can be explained that such OP of user U1depends on target rate R1. We can explain similarly for OP performance of other users at a high region of SNR. Figure 3 depicts similar trends of OP once we compare cases of target rates R1,R2and Rd. It can be seen that a lower required target rate results in better OP performance for these considered users. Fig. 2: Outage probability versus transmit SNR with different m. Fig. 3: Outage probability versus transmit SNR with different target rates. As presented in previous sections, Eq. (4), Eq. (5), Eq. (6), Eq. (7) and Eq. (8) mainly depend on power allocation factors, it is reported in terms of OP performance as Fig. 4. It is still seen that the OP of user U1is limited at high SNR. It can be concluded that by reconfiguration for power levels at relay R, we can change how good service to provide to users. The OP performance of different users in NOMA is remarkably improved at high SNR for user U1and D2D users. This is a promising result for designing NOMA in a practical scenario. These trends of OP for three users are similar to trends in Fig. 3. As a result, controlling such power factors lead to a crucial impact on performance gaps among three users in such NOMA system. It can be seen that how large amount of self-interference makes influence to outage performance of two users U1,U2as Fig. 5. It can be seen that high ξ3leads to worse OP performance. The main reason is that SINR is lower and the corresponding OP will be worse. Especially, ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 217 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER Tab. 1: All parameters in the related simulations. – Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 R1((bits·s−1)Hz−1) 1 – 1 1 1 R2((bits·s−1)Hz−1) 1 – 2 2 1 Rd((bits·s−1)Hz−1) 1 – 3 3 1 10.0820.0820.082–0.082 2111–1 30.120.120.12–0.12 m–322– α10.05 0.05 – 0.1 0.05 α20.05 0.05 – 0.1 0.05 α30.95 0.95 – 0.9 0.95 α40.95 0.95 – 0.9 0.95 λgU1←BS =λgD1←R0.01 0.01 0.01 0.01 0.01 λgR←BS =λgU1←R=λgU2←R0.01 0.01 0.01 0.01 0.01 λf0.01 0.01 0.01 0.01 0.01 such outage performance can be very bad in the case of ξ3= 1. Therefore, limiting the crucial impact of selfinterference is a way to improve the system performance. Fig. 4: Outage probability versus transmit SNR with different power allocation factors. Fig. 5: Outage probability versus transmit SNR with different self-interference levels. In Fig. 6, we simulate the throughput performance of the proposed scheme versus the transmit SNR at the BS. As shown in previous figures, the OP will be improved significantly at the high region of SNR. As a result, throughput can approach the ceiling value once SNR is greater than 30. By changing channel parameter m, just a slight change can be seen in these throughput curves. Throughput along with OP performance is helpful to give guidelines in the design of NOMA. Fig. 6: Throughput performance of U1,U2and D1. 5. Conclusion In this paper, we have studied a NOMA adopted at downlink to serve normal users and D2D users. In the proposed scheme, the fixed power allocation approach is adopted along with FD at the relay to improve spectrum efficiency. We considered this meaningful framework to look at outage and throughput performance of many kinds of users, and then employed the NOMA technique to enable the downlink signal processing. For the performance comparison 218 ©2021 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES VOLUME: 19 |NUMBER: 3 |2021 |SEPTEMBER on these users, we provided a comprehensive analysis of the outage behavior and derived the closed-form expressions of the outage probability. In the following, we consider the different performances of these users. A Nakagami-mfading model was employed to further provide a general case of NOMA. Numerical results are presented to verify the analysis in terms of the outage and throughput performance. Author Contributions T.-T.T.N. conceived of the presented idea, developed the theory and performed the computations, developed the theory and performed the computations. D.-T.D. encouraged T.-T.T.N. to investigate [aspecific aspect] and supervised the findings of this work. All authors discussed the results and contributed to the final manuscript. T.-T.T.N. carried out the experiment. D.-T.D. wrote the manuscript with support from A. T.-T.T.N. developed the theoretical formalism, performed the analytic calculations and performed the numerical simulations. Both T.-T.T.N. and D.-T.D. authors contributed to the final version of the manuscript. B. supervised the project. T.-T.T.N. planned and carried out the simulations. T.-T.T.N. and D.-T.D. contributed to the interpretation of the results. B. took the lead in writing the manuscript. All authors provided critical feedback and helped shape the research, analysis and manuscript. T.-T.T.N. designed the model and the computational framework and analysed the data and carried out the implementation and performed the calculations. D.-T.D. wrote the manuscript with input from T.-T.T.N. 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